The half-life of a certain tranquilizer in the bloodstream is 30 hours. How long will it take for the drug to decay to $85 \%$ of the original dosage? Use the exponential decay model, $A=A_{0} e^{k t}$, to solve.
hours
(Round to one decimal place as needed.)
Answer
Rounding to one decimal place, we find that it will take approximately \(\boxed{7.0}\) hours for the drug to decay to 85% of the original dosage.
Step 1 ：We are given that the half-life of a certain tranquilizer in the bloodstream is 30 hours. This means that after 30 hours, half of the tranquilizer will have decayed. We can use this information to find the decay constant $k$ in the exponential decay model $A=A_{0} e^{k t}$, where $A$ is the amount of the substance at time $t$, $A_{0}$ is the initial amount of the substance, and $t$ is the time.
Step 2 ：We set up the equation $0.5 = e^{30k}$ and solve for $k$. This gives us $k = -0.023104906018664842$.
Step 3 ：We are asked to find out how long it will take for the drug to decay to 85% of the original dosage. We can use the decay constant $k$ we just found to set up the equation $0.85 = e^{kt}$ and solve for $t$.
Step 4 ：Solving for $t$ gives us $t = 7.03395760911069$ hours.
Step 5 ：Rounding to one decimal place, we find that it will take approximately \(\boxed{7.0}\) hours for the drug to decay to 85% of the original dosage.
